Harry realized that he had just Apparated for the first time in his life.

This appears to settle the question of whether the school roof incident
mentioned in PS was a flying/levitation spell or Apparition. Levitation
spells and such are only supposed to work on humans up to a height of
about 5 feet or so (QA1),
and humans can't "fly" unaided at all
(QA1), but limited levitation is possible.

Well, here we go.

The first firsthand description of what Apparition is like.

It appears that he is now employing Occlumency against you.

So Voldemort knows that Occlumency exists and is himself an Occlumens.

...but we are, once again, one member of staff short.

I must say I (MLW) was completely suckered by this on first reading.
Dumbledore tells no lies here.

The odd chill that had lain over Privet Drive for two weeks persisted here too.

Since
we've seen similar remarks about a chill being put down to the presence
of dementors, and since Harry can see them if they're present, this is
an interesting remark.

You have not asked me, for instance, what is my favourite flavour of jam

Raspberry, as we learn shortly.

...although of course, if I were a Death Eater, I would have been
sure to research my own jam preferences before impersonating myself.

Like guessing people's passwords.

Inferi

As Dumbledore says, "They are corpses. Dead bodies that have been
bewitched to do a Dark wizard's bidding. Inferi have not been seen for
a long time, however, not since Voldemort was powerful."

He killed enough people to make an army of them, of course.

This suggests that to make an Inferius, the Dark wizard must have
killed the person contributing the dead body.

Ambrosius Flume, of Honeydukes

Ambrosia is a kind of dessert, which in turn is named for the food
of the Greek gods.

I think I know a lost cause when I see one.

I *love* watching Dumbledore yank Slughorn's chain like this.

I gather that you have been taking the Daily Prophet over the last two weeks?

This remark suggests that two weeks have elapsed since Harry's return to
Privet Drive at the end of last term.

"You have not asked me, for instance, what is my favorite
flavor of jam, to check that I am indeed Professor Dumbledore and not
an imposter. For future reference, Harry, it is raspberry...
although of course, if I were a Death Eater, I would have been sure to
research my own jam preferences before impersonating myself."

He seemed remarkably unabashed for a man who had just been
discovered pretending to be an armchair.

"It's quite easy once you know how, one simple Freezing
Charm on these absurd burglar alarms they use instead of Sneakoscopes
and make sure the neighbors don't spot you bringing in the piano."

"I take my hat off to you - or I would, if I were not afraid of
showering you in spiders."

We know the chapter begins on a Friday, because that's what Dumbledore
told Harry in his letter in HBP03.
The clock strikes midnight mid-chapter, after which Dumbledore tells
Harry that O.W.L. results are due that day. Since McGonagall said
in OP31 that O.W.L.s would be arriving
in July, Ron says in the next chapter
that it's "been a month" since O.W.L.s (which probably ended
on 18 June, also in OP31),
HBP06 says
that Harry was at the Burrow for 'a few weeks' prior to his
birthday, and we have a date reference because we know 1 March the next spring is a Saturday (HBP18),
it makes the most sense for this chapter to take place on 12-13 July.
Whew!